Brief descriptions and examples - statistical terms

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a) F distribution:
The F-distribution becomes relevant when we try to calculate the ratios of variances of normally distributed statistics. Suppose we have two samples with n1 and n2 observations, the ratio

is distributed according to an F distribution (named after R.A. Fisher) with df1 = n1-1 numerator degrees of freedom, and df2 = n2-1 denominator degrees of freedom. The F-distribution is skewed to the right, and the F-values can be only positive.

Example:
Tobacco and Alcohol Use in Animated Children's Movies:

Lists times (in seconds) that animated children's movies show tobacco use and alcohol use. The 50 times of tobacco use have a mean of 57.4 sec and a standard deviation of 104.0 sec. the 50 times of alcohol use have a mean of 32.46 sec and a standard deviation of 66.3 sec.

a. Assuming that we want to use the methods of this section to test the claim that the times of tobacco use and the times of alcohol use have different standards of deviations, identify F test statistic, critical value, and conclusion. Use a 0.05 significance level.

Step 1:
Null hypothesis:

H0: There is no significant difference in the minutes tobacco use and alcohol use.
Alternate hypothesis:

Ha: There is a significant difference in the minutes tobacco use and alcohol use.
Step 2:
Level of significance α=0.05 and critical value
Step 3:
Test statistics

Step 4:
Since the calculated value is greater than the critical value we reject the null hypothesis

Step 5:
Since Thus we conclude there is a significant difference in the minutes tobacco use and alcohol use i.e

b) F statistic
If X and Y are two independent chi-square variants with V1 and V2 d.f. respectively. Then F-statistic is defined by

In other words, F is defined as the ratio of two independent chi-square variants divided by their respective degrees of freedom and it follows snedecor's F- distribution with (v1,v2) d.f with probability function given by

c) Chi-square distribution
Chi -square test is a test which describes the magnitude of difference between observed frequencies and the frequencies expected under certain assumptions with the help of chi square , it is possible to find out whether such difference are significant or are insignificant and could have arisen due to fluctuations of sampling. In the chi-square test the only problem is to decide how the expected frequencies have to be arrived. There is no hard and fast rule for it and the method of arriving at the expected frequencies would depend on the nature of the problem.
Example:
1. Chi square analysis - Do the top 50 hitters come predominantly from one or several ...